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We prove that arbitrary homomorphisms from one of the groups      $$Homeo(2^\mathbb{N} ), Homeo(2^\mathbb{N} )^\mathbb{N} , Aut(\mathbb{Q}, < ), Homeo(\mathbb{R}) or Homeo(S^1 )$$       into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result of V. G. Pestov, that any action of the discrete group Homeo+(ℝ) by homeomorphisms on a compact metric space has a fixed point.

Automatic continuity of homomorphisms and fixed points on metric compacta